The Method of Fundamental Solutions for Inverse Problems Associated with the Steady-State Heat Conduction in the Presence of Sources
Liviu Marin

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 30, No. 2, pp. 99-122, 2008
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Keywords Meshless method, method of fundamental solutions, steady-state heat conduction, inverse problem, regularization, singular value decomposition.
Abstract The application of the method of fundamental solutions (MFS) to inverse boundary value problems associated with the steady-state heat conduction in isotropic media in the presence of sources, i.e. the Poisson equation, is investigated in this paper. Based on the approach of Alves and Chen\nobreakspace {}(2005), these problems are solved in two steps, namely by finding first an approximate particular solution of the Poisson equation and then the numerical solution of the resulting inverse boundary value problem for the Laplace equation. The resulting MFS discretised system of equations is ill-conditioned and hence it is solved by employing the singular value decomposition (SVD), whilst the choice of the optimal truncation number, which is the regularization parameter in this case, is based on the L-curve criterion. Three examples in smooth and piecewise smooth, simply and doubly connected, two-dimensional domains are considered and the convergence and stability of the proposed numerical method are analysed, based on the numerical experiments undertaken.
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